Tight Triangulation

A tight triangulation over a field is a connected simplicial complex triangulating a space such that every induced subcomplex has inclusion-induced maps

that are injective for every vertex subset and every homology degree . Thus no nonzero homology class supported on an induced subcomplex becomes zero when the remaining vertices and faces are added. Tightness can depend on the coefficient field (Kühnel and Lutz 2000, Bagchi et al. 2016).

The degree-zero condition implies neighborliness. Indeed, two nonadjacent vertices induce a disconnected pair whose two components become connected in . For closed surfaces, tightness over a field of characteristic 2 is equivalent to neighborliness. Over a field of characteristic different from 2 it is equivalent to neighborliness together with orientability (Bagchi et al. 2016).

Consequently, the Császár polyhedron gives a tight triangulation of the torus over every field. Its face numbers are and its edge graph is . This is a combinatorial statement about the triangulation, not an assertion that every geometric realization has the same extrinsic tightness properties.

Higher-dimensional examples include Kühnel's nine-vertex triangulation of the complex projective plane, which contains every possible edge and triangle. Kühnel's four-dimensional analogue of the Császár torus has 11 vertices and triangulates , with edge graph (Kühnel 1986, Kühnel and Lutz 2000). A complete edge graph alone does not imply tightness in higher dimensions.